Systems of Equations Unit

Multiple Choice

Warm Up!

After converting from standard form to slope-intercept form, what will the following equation look like?

$$4x+2y=6$$  

• $$y=2x+3$$  

• $$y=-\frac{1}{2}x-3$$  

• $$y=-2x+3$$  

• $$y=-\frac{1}{2}x+3$$  

Multiple Select

What is a solution to a system of linear equations?

• The point where both lines intersect

• The point where the y-axis and x-axis meet

• An ordered pair that both lines share

• A salt dissolved in water

Multiple Choice

When would we get "infinitely many solutions" to a system of equations?

• The lines are parallel, never intersecting

• The lines cross at one point

• The lines are the same and overlap at all points

• The lines cross at 2 points

Multiple Choice

When will we get "no solutions" from a system of equations?

• The two lines cross once

• The two lines are the same and overlap

• The two lines are parallel and never touch

• The two lines cross at 2 points

Multiple Choice

Which of the following graphs would fit the system of equations?

$$y=2x-4$$  

$$y=-\frac{1}{2}x+1$$  

Multiple Choice

Which point satisfies both equations?

y - x = 2

y + 2x = 2

• (0, 2)

• (2, 0)

• (2, 4)

• (-2, 6)

Multiple Choice

A jar contains only nickels and quarters. The total number of coins in the jar is 15. The total value of the coins is $2.35. Which system of equations represents this situation?

• n + q = 2.35

  .05n + .25q = 15

• n + q = 15

  .5n + .25q = 15

• n + q =15

  .05n + .25q = 2.35

• .05n + q = 15

  n + .25q = 2.35

Multiple Choice

The solution (x, y) to a system of equations is the point where they...? 

• Run off the graph

• Don't touch

• Intersect

• Exist

Multiple Choice

What is the proper way to set up the equation to begin solving if using the substitution method?

y = 6x − 11

and −2x − 3y = −7

• -2(-11) -3y= -7

• -2(6x-11) -3y = -7

• -2x -3(11) = -7

• -2x - 3(6x-11)= -7

Multiple Choice

If solving by elimination, what variable would be the BEST to eliminate?

4x + 8y = 20

and −4x + 2y = −30

• X because they have opposite coefficients

• Y because they have opposite coefficients

Multiple Choice

I can combine 2 linear equations if the coefficients are not opposites.

• True

• False

Multiple Choice

Let's multiply the top equation by -2. What would be our new equation?

2x + 3y = 12

4x - 7y = - 54

• 4x - 6y = -24

• -4x - 6y = 12

• -4x - 6y = 24

• -4x - 6y = -24

Poll

For the following equations, which coefficients would be easier to make opposites?

2x + 3y = 12

4x - 7y = - 54

• 2x and 4x

• 3y and -7y

Multiple Choice

If a system of linear equations has infinitely many solutions, what does this mean about the two lines? 

• Intersecting lines

• Same line

• Parallel lines

Multiple Choice

How can you tell if a point is a solution to a system?

• It makes the first equation true.

• The (x,y) coordinates satisfy both equations

• It makes logical sense

• It makes neither equation negative

Multiple Choice

If a system of equations has no solution, what does the graph look like? 

• intersecting lines

• parallel lines

• skew lines

• intersecting lines

Multiple Choice

If a system of linear equations has one solution, what does this mean about the two lines? 

• Parallel lines

• the same line 

• Intersecting lines

Fill in the Blanks

Multiple Choice

What is the solution to the systems of equations shown in the graph?

• 1 Solution x=-1/2

• 1 Solution y=-1/2

• Many Solutions

• No Solution

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Multiple Select

Solve the systems with any method you choose: Check all that apply:

• x= -5/2

• y=-5/2

• No Solutions

• Infinitely Many Solutions

• x=0

Multiple Select

Solve the systems with any method you choose: Check all that apply:

• x= -5/2

• y=-5/2

• No Solutions

• Infinitely Many Solutions

• x=0

Multiple Choice

David is running a concession stand at a soccer game. He sells nachos and sodas. Nachos cost $1.50 each and sodas cost $0.50 each. At the end of the game, David made a total of $78.50 and sold a total of 87 nachos and sodas combined. Which system of equations represents this situation?

• 1.5x+0.5y=78.5 x+y=87

• 1.5x+0.5y=78.5 1.5x+0.5y=87

• x+y=78.5 1.5x+0.5y=87

• x+y=78.5 x+y=87

Multiple Choice

Alexandra finds that she can give 3 haircuts and 2 hair dyes in 315 minutes. Giving 2 haircuts and 4 hair dyes takes 450 minutes.  Which system of equations represents the situation?

• 3x + 2y = 315 2x + 4y = 450

• 3x + 2y = 450 2x + 4y = 315

• 2x + 2y = 315 3x + 4y = 450

Multiple Choice

Is the ordered pair (7,-1) a solution of the following linear system?

• yes

• no

Multiple Choice

Which graph below shows the following system of equations? $$y=x\ +\ 7$$  

$$y=-\frac{1}{3}x+3$$  

Multiple Choice

Two high schools planned senior trips. High school A rented 1 van and 6 buses. They took 372 students. High school B rented 4 vans and 12 buses. They took 780 students. How many students does a van and bus hold?

• A van holds 18 students and a bus holds 59 students.

• A van holds 59 students and a bus holds 18 students.

• A van holds 24 students and a bus hold 54 students.

• A van holds 54 students and a bus holds 24 students.

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More money, more problems

​You recently received some birthday money, $130 to be exact. You count the number of bills you have and find that you have 17 bills, all either $10 or $5 bills.

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How many of each bill do you have?​

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Hint: Just try guessing and checking first!​

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How does this have to do with math?

Sometimes, in math, we are asked for two answers, like how many $5 bills AND $10 bills. Systems of equations are how we solve these problems.

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You've lost me...

Systems of equations are two or more equations that have the same variables like:​

y=x+1

y=-2x+6​

which both have x and y​

Systems of Equations

Systems of equations are used when we are given two pieces of information and asked for two answers from it, like our first question.

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Solutions are where two lines intersect on a graph and these are our answers.

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The solution here is (-1,3) because that is where the lines intersect!​

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​Solutions=answers

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Systems Intervention

Work on the following problems to further gauge your understanding of Systems of Equations.

Multiple Choice

Examine the student work shown. What mistake did the student make in using substitution to solve the system of equations?

• They isolated the variable incorrectly

• They distributed incorrectly

• They combined like terms incorrectly

• They used opposite operations incorrectly

• The number by itself represents (b) or the y-intercept.

• The number next to represents (m) or the slope.

• Make sure the equation is solve for y before determining the slope (m) and y-intercept (b).

Multiple Choice

Based on the equation, what is the slope (m)?

$$y=\frac{3}{4}x+10$$  

• $$4$$  

• $$3$$  

• $$10$$  

• $$\frac{3}{4}$$  

Multiple Choice

Based on the equation, what is the slope (m)?

$$y=-6$$  

• -6

• 0

• 1

• 6

Multiple Choice

Based on the equation, what is the slope (m)?

$$x=2$$  

• 2

• -2

• 0

• Undefined

Multiple Choice

Based on the equation, what is the slope (m)?

$$12x\ +4y\ =\ -60$$  

• -12

• 4

• -3

• 12

Multiple Choice

Based on the equation, what is the slope (m)?

$$10x-4y=-12$$  

• $$-\frac{5}{2}$$  

• $$-\frac{10}{4}$$  

• $$\frac{5}{2}$$  

• $$3$$  

One solution to a systems of equation problem is intersecting.

• Look at the the next few slides and determine which graph or graphs are intersecting.

• There may be more than one correct answer for each slide.

Multiple Select

Which graph or graphs shows a solution of intersecting?

Multiple Select

Which graph or graphs shows a solution of intersecting?

One solution to a systems of equation problem is No Solution.

• Look at the the next few slides and determine which graph or graphs are NO SOLUTION.

• There may be more than correct answer for each slide.

Multiple Select

Which graph or graphs shows a solution of NO SOLUTION?

Multiple Select

Which graph or graphs shows a solution of NO SOLUTION?

Sometimes equations are not in slope intercept form.

Take the following equations and put them into slope intercept form.

-Solve for y-

Multiple Choice

Put the following problem in slope intercept form.

$$6x+3y=0$$  

• $$y=\frac{1}{2}x$$  

• $$y=2x$$  

• $$y=-\frac{1}{2}x$$  

• $$y=-2x$$  

Multiple Choice

$$8x-5y=-10$$  Put the following problem in slope intercept form.

• $$y=-\frac{8}{5}x+2$$  

• $$y=\frac{8}{5}x+2$$  

• $$y=\frac{8}{5}x+10$$  

• $$y=-\frac{8}{5}x-2$$  

Multiple Choice

Put the following problem in slope intercept form.

$$y+3=-2\left(x-2\right)$$  

• $$y=-2x+1$$  

• $$y=-2x-7$$   

• $$y=2x+1$$  
  

• $$y=2x+7$$   

Poll

What would you like more help with? Select all that apply

• Graphing equations in slope intercept form

• Plotting Points

• Solving equations for y

• Understanding what solutions are possible in this unit

• Graphing equations NOT in slope intercept form